Digital compression has become ubiquitous and has been used in a wide variety of applications (such as video and audio applications). When looking to image capture (i.e., photography) as an example, an image sensor (i.e., charged-coupled device or CCD) is employed to generate analog image data, and an ADC is used to convert this analog image to a digital representation. This type of digital representation (which is raw data) can consume a huge amount of storage space, so an algorithm is employed to compress the raw (digital) image into a more compact format (i.e., Joint Photographic Experts Group or JPEG). By performing the compression after the image has been captured and converted to a digital representation, energy (i.e., battery life) is wasted. This type of loss is true for nearly every application in which data compression is employed.
Compressive sensing is an emerging field that attempts to prevent the losses associated with data compression and improve efficiency overall. Compressive sensing looks to perform the compression before or during capture, before energy is wasted. To accomplish this, one should look to adjusting the theory under which the ADCs operate, since the majority of the losses are due to the data conversion. For ADCs to perform properly under conventional theories, the ADCs should sample at twice the highest rate of the analog input signal (i.e., audio signal), which is commonly referred to as the Shannon-Nyquist rate or Nyquist frequency. Compressive sensing should allow for a sampling rate well-below the Shannon-Nyquist rate so long as the signal of interest is sparse in some arbitrary representing domain and sampled or sensed in a domain which is incoherent with respect to the representation domain.
As is apparent, a portion of compressive sensing is devoted to reconstruction (usually in the digital domain) after resolution; an example of which is described below with respect to a successive approximation register (SAR) ADC and in Luo et al., “Compressive Sensing with a Successive Approximation ADC Architecture,” 2011 Intl. Conf on Acoustic Speech and Signal Processing (ICASSP), pp 2590-2593. For the compressive sensing framework, a signal {right arrow over (y)} can be expressed as:{right arrow over (y)}= Φ Ψ{right arrow over (α)}= A{right arrow over (α)},  (1)where {right arrow over (a)} (which satisfies the condition {right arrow over (a)}εN) is a frequency sparse signal, Ψ is the sparsifying basis matrix, Φ is a row restriction of the identity matrix that provides M samples from a random set Ω (or Φ= I|ΩεM×N), and A is a measurement matrix. The measurement matrix A should obey the restricted isometry property (RIP) with high probability as long as the number of measurements or samples M is sufficiently large.
As is apparent from equation (1), the reconstruction is based on an accurate sparsifying basis Ψ; any mismatch from this basis limits reconstruction performance significantly. As an example, it can be assumed that basis Ψ is an inverse fast Fourier transform or IFFT matrix (which would map frequency sparse signal {right arrow over (α)} to the time domain). For this example, basis mismatch occurs as spectral leakage when taking random Fourier measurements. Looking to the example in FIG. 1, a discrete complex sinusoid lies at an integer frequency k is shown, and its spectrum consists of a single tone that lies on an inverse discrete Fourier Transform (IDFT) bin, indicating small to no mismatch. Looking to FIG. 2, on the other hand, the frequency discrete complex sinusoid is offset by one-half of a IDFT bin, and this causes significant spectral leakage due to the model mismatch, even though discrete complex sinusoid has a single tone.
Thus, there is a need for a method and/or apparatus that compensates for sparsifying basis mismatch.
Some conventional circuits and systems are: U.S. Pat. No. 7,324,036; U.S. Pat. No. 7,834,795; Luo et al., “Compressive Sensing with a Successive Approximation ADC Architecture,” 2011 Intl. Conf on Acoustic Speech and Signal Processing (ICASSP), pp 2590-2593; R. Baraniuk, “Compressive sensing,” Lecture notes in IEEE Signal Processing magazine, 24(4):118-120, 2007; Candes et al., “Compressed sensing with coherent and redundant dictionaries,” Applied and Computational Harmonic Analysis, 2010; Duarte et al., “Spectral compressive sensing,” 2010; Eldar et al. “Compressed sensing for analog signals,” IEEE Trans. Signal Proc., 2008, submitted; Mishali et al. “Blind multi-band signal reconstruction: Compressed sensing for analog signals,” IEEE Trans. Signal Proc., 2007, submitted; Rudelson et al., “On sparse reconstruction from fourier and gaussian measurements,” Communications on Pure and Applied Mathematics, 61(8):1025-1045, 2008; Tropp et al., “Signal recovery from partial information via orthogonal matching pursuit,” IEEE Trans. Info. Theory, 53(12):4655-4666, December 2007; Tropp et al., “Random_lters for compressive sampling and reconstruction,” In IEEE Int. Conf on Acoustics, Speech and Signal Processing (ICASSP), volume III, pages 872-875, Toulouse, France, May 2006, submitted; Tropp et al., “Beyond Nyquist: E_cient sampling of sparse bandlimited signals” 2009 Preprint; van den Berg et al., “SPGL1: A solver for large-scale sparse reconstruction,” June 2007, http://www.cs.ubc.ca/labs/scl/spgl1; and van den Berg et al. “Probing the pareto frontier for basis pursuit solutions,” SIAM Journal on Scientific Computing, 31(2):890-912, 2008.